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Mirrors > Home > MPE Home > Th. List > mircgrextend | Structured version Visualization version GIF version |
Description: Link congruence over a pair of mirror points. cf tgcgrextend 26576. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mirtrcgr.e | ⊢ ∼ = (cgrG‘𝐺) |
mirtrcgr.m | ⊢ 𝑀 = (𝑆‘𝐵) |
mirtrcgr.n | ⊢ 𝑁 = (𝑆‘𝑌) |
mirtrcgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirtrcgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirtrcgr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
mirtrcgr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
mircgrextend.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) |
Ref | Expression |
---|---|
mircgrextend | ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | mirtrcgr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | mirtrcgr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mirtrcgr.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐵) | |
10 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircl 26752 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
11 | mirtrcgr.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
12 | mirtrcgr.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
13 | mirtrcgr.n | . . 3 ⊢ 𝑁 = (𝑆‘𝑌) | |
14 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircl 26752 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃) |
15 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mirbtwn 26749 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴)) |
16 | 1, 2, 3, 4, 10, 6, 5, 15 | tgbtwncom 26579 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴))) |
17 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mirbtwn 26749 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋)) |
18 | 1, 2, 3, 4, 14, 12, 11, 17 | tgbtwncom 26579 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋))) |
19 | mircgrextend.1 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) | |
20 | 1, 2, 3, 4, 5, 6, 11, 12, 19 | tgcgrcomlr 26571 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋)) |
21 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircgr 26748 | . . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴)) |
22 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircgr 26748 | . . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋)) |
23 | 20, 21, 22 | 3eqtr4d 2787 | . 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋))) |
24 | 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23 | tgcgrextend 26576 | 1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 distcds 16811 TarskiGcstrkg 26521 Itvcitv 26527 LineGclng 26528 cgrGccgrg 26601 pInvGcmir 26743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-trkgc 26539 df-trkgb 26540 df-trkgcb 26541 df-trkg 26544 df-mir 26744 |
This theorem is referenced by: mirtrcgr 26774 |
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